A variety of analytical approaches have been developed for the study of quantum spin systems in two dimensions, the notable ones being spin-waves, slave boson/fermion parton constructions, and for lattices with one-to-one local correspondence of faces and vertices, the 2D Jordan-Wigner (JW) fermionization. Field-theoretically, JW fermionization is implemented through Chern-Simons (CS) flux attachment. For a correct fermionization of lattice quantum spin-$1/2$ magnets, it is necessary that the fermions obey mutual bosonic (anyonic) statistics under exchange - this is not possible to implement on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge fields. Enlarging the gauge degrees of freedom to include the dual lattice allows the construction of consistent mutual Chern-Simons field theories. Here we propose a mutual CS theory where the microscopic (spin) degrees of freedom are represented as lattice fermionic matter additionally coupled to specific combinations of dual lattice gauge fields that depend on the local geometry. We illustrate the use of this method for understanding the properties of a honeycomb Kitaev model subjected to a strong Zeeman field in the $z$-direction. Our CS gauge theory framework provides an understanding why the topological phase is degraded at lower (higher) critical fields for the ferro- (antiferro-) magnetic Kitaev interaction. Additionally, we observe an effectively one-dimensional character of the low-excitations at higher fields in the $z$-direction which we also confirm by spin-wave calculations.
Fermionization Transform for Certain Higher-Dimensional Quantum Spin Models
提出了一种推广的Jordan-Wigner变换，它允许精确地费米子化一个大家族的量子自旋哈密顿量在维度高于一。关键的新步骤是通过向原始模型添加一组独立的自由自旋来扩大原始模型的希尔伯特空间，并使用这些辅助算子和晶格自旋的组合来构建物理哈密顿量的适当费米子表示。该变换对于晶格自旋哈密顿量特别有用，其中XY型的双自旋相互作用不存在或仅存在于一维链中，并且其中链通过伊辛型的双自旋相互作用、环交换项或更一般的多自旋相互作用耦合，所述多自旋相互作用涉及来自每个链的偶数个自旋算子。使用所提出的费米子化方法，我们提供了一个简单的参数，表明与Paramekanti，Balents和Fisher [Phys. Rev. B 66，054526（2002）]提出的环交换模型密切相关的自旋哈密顿量确实实现了自旋液体状态。
Properties and application of the SO(3) Majorana representation of spin: equivalence with the Jordan-Wigner transformation and exact $Z_{2}$ gauge theories for spin models
